Given plane Π : f(x,y,z) = 4x+3y-z = -1
Need to find point P on Π that is closest to the origin O=(0,0,0).
Solution:
First step: check if O is on the plane Π : f(0,0,0)=0 ≠ -1 => O is not on Π
Next:
We know that the required point must lie on the normal vector <4,3,-1> passing through the origin, i.e.
P=(0,0,0)+k<4,3,-1> = (4k,3k,-k)
For P to lie on plane Π , it must satisfy
4(4k)+3(3k)-(-k)=-1
Solving for k
k=-1/26
=>
Point P is (4k,3k,-k) = (-4/26, -3/26, 1/26) = (-2/13, -3/26, 1/26)
because P is on the normal vector originating from the origin, and it satisfies the equation of plane Π
Answer: P(-2/13, -3/26, 1/26) is the point on Π closest to the origin.
I was stuck on the same thing in my class test. I ended up failing but if I get the answers to it I’ll totally send them to you
A second degree polynomial function has the general form:
, where
.
The leading coefficient is a, so we have a=-1.
5 is a double root means that :
i) f(5)=0,
ii) the discriminant D is 0, where
.
Substituting x=5, we have
f(5)=a(5)^2+b(5)+c,
and since f(5)=0, and a is -1 we have:
0=-25+5b+c
thus c=25-5b.
By ii)
.
Substituting a with -1 and c with 25-5b we have:
Finally we find c: c=25-5b=25-50=-25
Thus the function is
Remark: It is also possible to solve the problem by considering the form
directly.
In general, if a quadratic function has leading coefficient a, and has a root r of multiplicity 2, then its form is
#66 Dimensions of matrix: The dimensions of a matrix are
[number of rows (across) x number of columns (up and down)]
There is 1 row and 4 columns, hence the dimensions of [1 x 4]
#71: use factor/label conversion method.
480 minutes 1 hour 480 hours 8 hours
* -------------- = -------------- =
60 minutes 60 minutes
#72: common difference is how you get from one number to the next in the sequence. So from 42 to 38, you subtract 4, and the pattern is subtract 4 for the whole sequence. So the common difference <em></em>is -4
#38: The midpoint is the middle of the two coordinates, so imagine when used to find mean by drawing a number line like 1 to 10 and the mean would be 5. What did you do? You added 1 and 10 and divided by 2.
You can use the same method for finding midpoints on the coordinate plane.
The formula for midpoint is <span><span>(<span><span>x1+x</span>2</span>)</span>,<span>(<span><span>y1+y</span>2</span>). So it's (5 + 2) , (8 + 7) which equals (7, 15).
#35: This is the same 3* x^(9*1/5) = 3*x^9/5
Remember square root of a number is the same as that number to the 1/2, and the same with all other roots. The fifth root of x is the same as x to the power of 1/5
#36: This is the same as 36*( x^5*3/2 * y^3*3/2), since you can put the power to each of the numbers almost like the distributive property.
= 36 * (x^15/2 * y^9/2)
Well #20 is <em>extremely</em><em> </em>easy. The solutions are anything that is shaded in the blue
The rightmost question is the same thing as #20. Anything not shaded is not a solution.
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#64: Just like you learned in elementary school, the median is the middle number. The middle number is 10<span>
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